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The Cartesian subtangent is defined as the projection of part of tangent intercepted between the point of contact and the $X$-axis on the $X$-axis; and similarly, the subnormal is the projection of normal on $X$-axis.

But in polar coordinate system subtangent is defined differently and so is subnormal! But I think these two definition of subtangent and subnormal are not consistent.

Shouldn't polar subtangent similarly be defined as the projection of tangent on initial line and so should subtangent defined?

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In polar coordinates, you are defining the polar subtangent, not the cartesian subtangent. Why should the "polar subtangent" be the same as the "cartesian subtangent"? Is Jacob Bernoulli the same person as Daniel Bernoulli? No, the two definitions are not consistent, since the cartesian subtangent is always horizontal, but the polar subtangent is not. So what? –  Arturo Magidin Jun 22 '12 at 3:52
A circle is always a circle; whether it is in cartesian coordinate system or in polar system. Even tangents are also same in both the coordinate systems. But how could sub tangents and sub normals be different? Should then subtangent and subnormal be understood as just co-incidence of bearing same name but completely different things? –  Prakash Gautam Jun 22 '12 at 6:02
A "circle" is always a circle. But why should a "cartesian subtangent" be the same thing as a "polar subtangent"? Just because you decide to drop the "cartesian" and the "polar" adjective doesn't mean that they are the same thing. Hence my analogy to the Bernoullis. You are just looking at the last name, and ignoring the first names. No, they are not "completely different things": they are similar things, defined in terms of similar things, but they are not identical, just like Jacob and Daniel Bernoulli are related, but not identical. –  Arturo Magidin Jun 22 '12 at 14:39
Note also that a Euclidean circle is not the same as a "sup-norm" circle, for that matter... –  Arturo Magidin Jun 22 '12 at 17:17
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closed as not a real question by Pedro Tamaroff, William, rschwieb, Arkamis, Norbert Oct 13 '12 at 17:35

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1 Answer

Shouldn't polar subtangent similarly be defined as the projection of tangent on initial line and so should subtangent defined?

The answer is no. In polar coordinates $(r,\theta)$ we have a distinguished point, the origin, which corresponds to $r=0$. But the initial [half]-line $\theta=0$ carries no such significance. A helpful geographic analogy is: the North pole is objectively a special place on the Earth (so is the South pole, of course), but there is nothing special about the Greenwich meridian to distinguish it among other meridians.

Therefore, any reasonable geometric definition in polar coordinates should be independent of our (arbitrary) choice of initial mark $\theta=0$. The Wikipedia definition of polar subtangent and subnormal meets this criterion; yours does not.

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